This work focuses on the study of the dynamic behavior and lattice-size dependence of the surface root-mean-square slope of thin-film deposition processes that involve thermal balance between film growth and surface relaxation. Two different deposition processes taking place on square and triangular lattices are introduced and used to investigate the dynamics and lattice-size dependence of the surface root-mean-square slope. The simulation results indicate that the expected mean slope square reaches quickly a steady-state value and exhibits a very weak dependence with respect to lattice size variation. The simulation findings are corroborated by an analysis of appropriate finite-difference discretizations of surface height profiles computed by an Edwards-Wilkinson-type partial differential equation that can be used to describe the dynamics of surface height profile in the deposition processes under consideration.